What are Sin, Cos, and Tan? [pptime time="0:01"](0:01) [/pptime]

In a right-angled triangle, sine (sin), cosine (cos), and tangent (tan) are trigonometric ratios that compare two sides relative to a given angle θ : Sine (sin θ) is the ratio of the opposite side of angle θ to the hypotenuse. $$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$ Cosine (cos θ) is the ratio of the adjacent side of angle θ to the hypotenuse. $$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$ Tangent (tan θ) is the ratio of the opposite side of angle θ to the adjacent side of angle θ. $$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$$ 💡 SOH-CAH-TOA is an easy way to remember them! 🔑 Key Understanding: The values of sin θ, cos θ, and tan θ depend only on the angle θ , not the size of the triangle. Even if the triangle gets bigger or smaller, the ratios remain the same as long as the angle θ stays the same.

Trigonometry: Sine, Cosine, Tangent

🎬 Video Tutorial

What are Sin, Cos, and Tan? (0:01)

In a right-angled triangle, sine (sin), cosine (cos), and tangent (tan) are trigonometric ratios that compare two sides relative to a given angle θ:

Sine (sin θ) is the ratio of the opposite side of angle θ to the hypotenuse. $$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$
Cosine (cos θ) is the ratio of the adjacent side of angle θ to the hypotenuse. $$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$
Tangent (tan θ) is the ratio of the opposite side of angle θ to the adjacent side of angle θ. $$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$$

💡 SOH-CAH-TOA is an easy way to remember them!

🔑 Key Understanding: The values of sin θ, cos θ, and tan θ depend only on the angle θ, not the size of the triangle. Even if the triangle gets bigger or smaller, the ratios remain the same as long as the angle θ stays the same.

How to Use Sin to Find a Side? (0:56)

We can use sine to find a missing side in a right-angled triangle. If we are given:

Angle: $30^\circ$
Opposite side: $4 \text{ cm}$

To find the Hypotenuse, we use the sine formula: $\sin 30^\circ = \frac{4\text{ cm}}{\text{Hypotenuse}}$ Since $\sin 30^\circ = \large \frac{1}{2}$ $\text{Hypotenuse} = 4 \text{ cm} \times 2 = 8 \text{ cm}$

How to Use Tan to Find a Side? (1:40)

We can use tangent to find a missing side in a right-angled triangle. If we are given:

Angle: $50^\circ$
Adjacent side: $5 \text{ cm}$

To find the opposite side, we use the tangent formula: $$\tan 50^\circ = \frac{\text{Opposite}}{5 \text{ cm}}$$

Since $ \tan 50^\circ \approx 1.191753 $, we have:

$$\text{Opposite} \approx 5 \text{ cm} \times 1.191753 \approx 6 \text{ cm}$$

How to Use Cos to Find an Angle? (2:14)

We can use cosine to find a missing angle in a right-angled triangle. If we are given:

Adjacent side: $4 \text{ cm}$
Hypotenuse: $8 \text{ cm}$

To find the angle $\theta$, we use the cosine formula: $$\cos \theta = \frac{4 \text{ cm}}{8 \text{ cm}} = \frac{1}{2}$$

Now, solve for $\theta$ by using the inverse cosine function:

$$\theta = \cos^{-1} \left(\frac{1}{2} \right)$$

Now, solve for $\theta$ by using the inverse cosine function on a calculator:

Press the cos⁻¹ button (If you can’t find it, press SHIFT, then COS)
Enter $\large \frac{1}{2}$ into the calculator.
Press =, and the calculator will display 60°

Thus, the missing angle is: $$\theta = 60^\circ$$